Fit Procedure

Another noteworthy observation is that at both ends of the curve in figure 3.3 some scatter is present. This has no physical meaning, but results from the simple fact that a simulation is limited to a certain time span. The curve fit procedure for the monomer concentration versus time is therefore less accurate at both ends of the curve as compared to the intermediate part. Upon differentiation, this effect is aggravated and results in scatter in the kt data. These data will be omitted in the following graphs. 

---

Constants A, and c are estimated by the least squares fitting procedure. 

With XRD applied to bulk materials, a detailed structural analysis of atomic positions is rather straightforward and routine for structures that can be quite complex (see chapter B 1.9) direct methods in many cases give good results in a single step, while the resulting atomic positions may be refined by iterative fitting procedures based on simulation of the diffraction process. 

A molecular fitting algorithm requires a numerical measure of the difference between two structures when they are positioned in space. The objective of the fitting procedure is to find the relative orientations of the molecules in which this function is minimised. The most common measure of the fit between two structures is the root mean square distance between pairs of atoms, or RMSD ... 

If the mahix form of the fitting procedure is used to solve for the intercept and slope of a shaight line, Eq. (3-13)... 

Write the determinant for a sixth-degree curve fitting procedure. 

Table 6.1 gives the mathematical forms of energy terms often used in popular force fields. The constants may vary from one force field to another according to the designer s choice of unit system, zero of energy, and fitting procedure. 

The equation of Knox et al. was not derived theoretically from a basic dispersion model, but the constants of the equation were determined by a curve fitting procedure... 

The results obtained were probably as accurate and precise as any available and, consequently, were unique at the time of publication and probably unique even today. Data were reported for different columns, different mobile phases, packings of different particle size and for different solutes. Consequently, such data can be used in many ways to evaluate existing equations and also any developed in the future. For this reason, the full data are reproduced in Tables 1 and 2 in Appendix 1. It should be noted that in the curve fitting procedure, the true linear velocity calculated using the retention time of the totally excluded solute was employed. An example of an HETP curve obtained for benzyl acetate using 4.86%v/v ethyl acetate in hexane as the mobile phase and fitted to the Van Deemter equation is shown in Figure 1. 

Table 2. Experimental Values for the Dispersion Equation Coefficients Obtained by a Curve Fitting Procedure...

It is now necessary to identify the correct functions of the capacity factor (k") to be used in equation (12) to evaluate the diffusivity. Knox [7] suggested the following approach which required a polynomial curve fitting procedure to identify the necessary constants. Rearranging equation (12),... 

Other strength criteria are described by Sendeckyj [2-28]. Tennyson, MacDonald, and Nanyaro addressed the next logical step in a curve-fitting procedure, namely a third-order polynomial fit to failure data [2-29], However, the added complexity of their criterion has limited its use even though they identified some loading conditions under which their criterion is necessary to properly describe the actual failure behavior. 

Ri seems to be related to solute size, but it is obtained by an iterative curve-fitting procedure. The parameters M and D are characteristic of the solvent. For water Af = 0. A solvophobicity parameter Sp is then defined for the solvent having value M... 

Many of these fitting schemes were derived before linear scaling techniques (Section 3.8.6) were fully developed, and it is not clear whether they have any advantages. For calculation of energy derivatives, they acmally seem counterproductive, since the fitting procedures seriously complicate the computational expressions. ... 

FIGURE 11.10 Removal of outliers points to achieve curve fits, (a) The least squares fitting procedure cannot fit a sigmoidal curve to the data points due to the ordinate value at 20j.iM. Removal of this point allows an estimate of the curve, (b) The outlier point at 2jiM causes a capricious and obviously errant fit to the complete data set. Removal of this point indicates a clearer view of the relationship between concentration and response. 

FIGURE 11.12 Overextrapolation of data, (a) Nonlinear curve fitting techniques estimate an ordinate maximal asymptote that is nearly 100% beyond the last available data point, (b) The curve fitting procedure estimates a maximal asymptote much closer to the highest available data point. A useful rule is to reject fits that cause an estimated maximal asymptote that is >25% the value of the highest available data point. 

---